The objective of the project is to contribute (through the development of mathematical tools for the study of information theory in physics and statistics) to strategic research related to the problems of extraction, processing and protection of classical and quantum information. We shall deal with three different, but closely related topics.

The first group of problems concerns the Lempel–Ziv coding algorithm whose performance studies have led to deep insights into specific and relative entropies of stationary measures on shift spaces. Notable among those is the characterisation of the specific entropy of a stochastic source in terms of the asymptotics of recurrence times of a typical signal and the specific cross entropy in terms of waiting times. We shall investigate in depth the mathematical theory of entropic estimators with emphasis on their fluctuations and fractal dimension theory of the level sets.

The second group concerns bipartite systems with infinitely many degrees of freedom. For those systems, it is not clear what is a good measure of entanglement. Recently, much progress has been made in the setting of quantum fields on flat space, and computations in simple models are starting to reveal mechanisms responsible for geometric formulae for entanglement entropy. Furthermore, there are proposals on how to give meaning to information for subsystems such as a single wave. Our goal is to develop more realistic models in relativistic physics, with applications to QFT on AdS spaces and the black hole information paradox.

The third group deals with fluid flows in a pipe. An important question from the point of view of applications is the estimation of viscosity using long-time observations of the kinetic energy and vorticity. This requires investigation of probabilistic limit theorems for the observables and the expression of resulting quantities in terms of viscosity. Our goal is to find reliable estimators and to investigate their optimality.